2,072 research outputs found
Surrogate-based Ensemble Grouping Strategies for Embedded Sampling-based Uncertainty Quantification
The embedded ensemble propagation approach introduced in [49] has been
demonstrated to be a powerful means of reducing the computational cost of
sampling-based uncertainty quantification methods, particularly on emerging
computational architectures. A substantial challenge with this method however
is ensemble-divergence, whereby different samples within an ensemble choose
different code paths. This can reduce the effectiveness of the method and
increase computational cost. Therefore grouping samples together to minimize
this divergence is paramount in making the method effective for challenging
computational simulations. In this work, a new grouping approach based on a
surrogate for computational cost built up during the uncertainty propagation is
developed and applied to model diffusion problems where computational cost is
driven by the number of (preconditioned) linear solver iterations. The approach
is developed within the context of locally adaptive stochastic collocation
methods, where a surrogate for the number of linear solver iterations,
generated from previous levels of the adaptive grid generation, is used to
predict iterations for subsequent samples, and group them based on similar
numbers of iterations. The effectiveness of the method is demonstrated by
applying it to highly anisotropic diffusion problems with a wide variation in
solver iterations from sample to sample. It extends the parameter-based
grouping approach developed in [17] to more general problems without requiring
detailed knowledge of how the uncertain parameters affect the simulation's
cost, and is also less intrusive to the simulation code.Comment: 24 pages, 4 figure
A computational methodology for two-dimensional fluid flows
A weighted residual collocation methodology for simulating two-dimensional
shear-driven and natural convection flows has been presented. Using a dyadic
mesh refinement, the methodology generates a basis and a multiresolution scheme
to approximate a fluid flow. To extend the benefits of the dyadic mesh
refinement approach to the field of computational fluid dynamics, this article
has studied an iterative interpolation scheme for the construction and
differentiation of a basis function in a two-dimensional mesh that is a finite
collection of rectangular elements. We have verified that, on a given mesh, the
discretization error is controlled by the order of the basis function. The
potential of this novel technique has been demonstrated with some
representative examples of the Poisson equation. We have also verified the
technique with a dynamical core of two-dimensional flow in primitive variables.
An excellent result has been observed-on resolving a shear layer and on the
conservation of the potential and the kinetic energies with respect to
previously reported benchmark simulations. In particular, the shear-driven
simulation at CFL = 2.5 (Courant-Friedrichs-Lewy) and
(Reynolds number) exhibits a linear speedup of CPU time with an increase of the
time step, . For the natural convection flow, the conversion of the
potential energy to the kinetic energy and the conservation of total energy is
resolved by the proposed method. The computed streamlines and the velocity
fields have been demonstrated.Comment: 29 pages, 7 figure
Adaptive 2D IGA boundary element methods
We derive and discuss a posteriori error estimators for Galerkin and
collocation IGA boundary element methods for weakly-singular integral equations
of the first-kind in 2D. While recent own work considered the Faermann residual
error estimator for Galerkin IGA boundary element methods, the present work
focuses more on collocation and weighted- residual error estimators, which
provide reliable upper bounds for the energy error. Our analysis allows
piecewise smooth parametrizations of the boundary, local mesh-refinement, and
related standard piecewise polynomials as well as NURBS. We formulate an
adaptive algorithm which steers the local mesh-refinement and the multiplicity
of the knots. Numerical experiments show that the proposed adaptive strategy
leads to optimal convergence, and related IGA boundary element methods are
superior to standard boundary element methods with piecewise polynomials.Comment: arXiv admin note: text overlap with arXiv:1408.269
An efficient, globally convergent method for optimization under uncertainty using adaptive model reduction and sparse grids
This work introduces a new method to efficiently solve optimization problems
constrained by partial differential equations (PDEs) with uncertain
coefficients. The method leverages two sources of inexactness that trade
accuracy for speed: (1) stochastic collocation based on dimension-adaptive
sparse grids (SGs), which approximates the stochastic objective function with a
limited number of quadrature nodes, and (2) projection-based reduced-order
models (ROMs), which generate efficient approximations to PDE solutions. These
two sources of inexactness lead to inexact objective function and gradient
evaluations, which are managed by a trust-region method that guarantees global
convergence by adaptively refining the sparse grid and reduced-order model
until a proposed error indicator drops below a tolerance specified by
trust-region convergence theory. A key feature of the proposed method is that
the error indicator---which accounts for errors incurred by both the sparse
grid and reduced-order model---must be only an asymptotic error bound, i.e., a
bound that holds up to an arbitrary constant that need not be computed. This
enables the method to be applicable to a wide range of problems, including
those where sharp, computable error bounds are not available; this
distinguishes the proposed method from previous works. Numerical experiments
performed on a model problem from optimal flow control under uncertainty verify
global convergence of the method and demonstrate the method's ability to
outperform previously proposed alternatives.Comment: 27 pages, 6 figures, 1 tabl
An adaptive high order direct solution technique for elliptic boundary value problems
This manuscript presents an adaptive high order discretization technique for
elliptic boundary value problems. The technique is applied to an updated
version of the Hierarchical Poincar\'e-Steklov (HPS) method. Roughly speaking,
the HPS method is based on local pseudospectral discretizations glued together
with Poincar\'e-Steklov operators. The new version uses a modified tensor
product basis which is more efficient and stable than previous versions. The
adaptive technique exploits the tensor product nature of the basis functions to
create a criterion for determining which parts of the domain require additional
refinement. The resulting discretization achieves the user prescribed accuracy
and comes with an efficient direct solver. The direct solver increases the
range of applicability to time dependent problems where the cost of solving
elliptic problems previously limited the use of implicit time stepping schemes
The Isogeometric Nystr\"om Method
In this paper the isogeometric Nystr\"om method is presented. It's
outstanding features are: it allows the analysis of domains described by many
different geometrical mapping methods in computer aided geometric design and it
requires only pointwise function evaluations just like isogeometric collocation
methods. The analysis of the computational domain is carried out by means of
boundary integral equations, therefor only the boundary representation is
required. The method is thoroughly integrated into the isogeometric framework.
For example, the regularization of the arising singular integrals performed
with local correction as well as the interpolation of the pointwise existing
results are carried out by means of Bezier elements.
The presented isogeometric Nystr\"om method is applied to practical problems
solved by the Laplace and the Lame-Navier equation. Numerical tests show higher
order convergence in two and three dimensions. It is concluded that the
presented approach provides a simple and flexible alternative to currently used
methods for solving boundary integral equations, but has some limitations.Comment: 21 Figure
An Error Analysis of Some Higher Order Space-Time Moving Finite Elements
This is a study of certain finite element methods designed for
convection-dominated, time-dependent partial differential equations.
Specifically, we analyze high order space-time tensor product finite element
discretizations, used in a method of lines approach coupled with mesh
modification to solve linear partial differential equations. Mesh modification
can be both continuous (moving meshes) and discrete (static rezone). These
methods can lead to significant savings in computation costs for problems
having solutions that develop steep moving fronts or other localized
time-dependent features of interest. Our main result is a symmetric a priori
error estimate for the finite element solution computed in this setting
Simplex Stochastic Collocation for Piecewise Smooth Functions with Kinks
Most approximation methods in high dimensions exploit smoothness of the
function being approximated. These methods provide poor convergence results for
non-smooth functions with kinks. For example, such kinks can arise in the
uncertainty quantification of quantities of interest for gas networks. This is
due to the regulation of the gas flow, pressure, or temperature. But, one can
exploit that for each sample in the parameter space it is known if a regulator
was active or not, which can be obtained from the result of the corresponding
numerical solution. This information can be exploited in a stochastic
collocation method. We approximate the function separately on each smooth
region by polynomial interpolation and obtain an approximation to the kink.
Note that we do not need information about the exact location of kinks, but
only an indicator assigning each sample point to its smooth region. We obtain a
global order of convergence of , where is the degree of the
employed polynomials and the dimension of the parameter space
Fujiwhara interaction of tropical cyclone scale vortices using a weighted residual collocation method
The fundamental interaction between tropical cyclones was investigated
through a series of water tank experiements by Fujiwhara [20, 21, 22]. However,
a complete understanding of tropical cyclones remains an open research
challenge although there have been numerous investigations through measurments
with aircrafts/satellites, as well as with numerical simulations. This article
presents a computational model for simulating the interaction between cyclones.
The proposed numerical method is presented briefly, where the time integration
is performed by projecting the discrete system onto a Krylov subspace. The
method filters the large scale fluid dynamics using a multiresolution
approximation, and the unresolved dynamics is modeled with a Smagorinsky type
subgrid scale parameterization scheme. Numerical experiments with Fujiwhara
interactions are considered to verify modeling accuracy. An excellent agreement
between the present simulation and a reference simulation at Re = 5000 has been
demonstrated. At Re = 37440, the kinetic energy of cyclones is seen
consolidated into larger scales with concurrent enstrophy cascade, suggesting a
steady increase of energy containing scales, a phenomena that is typical in
two-dimensional turbulence theory. The primary results of this article suggest
a novel avenue for addressing some of the computational challenges of mesoscale
atmospheric circulations.Comment: 24 pages, 11 figures, submitte
Enhancing adaptive sparse grid approximations and improving refinement strategies using adjoint-based a posteriori error estimates
In this paper we present an algorithm for adaptive sparse grid approximations
of quantities of interest computed from discretized partial differential
equations. We use adjoint-based a posteriori error estimates of the physical
discretization error and the interpolation error in the sparse grid to enhance
the sparse grid approximation and to drive adaptivity of the sparse grid.
Utilizing these error estimates provides significantly more accurate functional
values for random samples of the sparse grid approximation. We also demonstrate
that alternative refinement strategies based upon a posteriori error estimates
can lead to further increases in accuracy in the approximation over traditional
hierarchical surplus based strategies. Throughout this paper we also provide
and test a framework for balancing the physical discretization error with the
stochastic interpolation error of the enhanced sparse grid approximation
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